Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{4k^2 - 44k + 72}{2k^2 + 8k + 6} \times \dfrac{4k + 4}{2k - 4} $
Answer: First factor out any common factors. $p = \dfrac{4(k^2 - 11k + 18)}{2(k^2 + 4k + 3)} \times \dfrac{4(k + 1)}{2(k - 2)} $ Then factor the quadratic expressions. $p = \dfrac {4(k - 2)(k - 9)} {2(k + 1)(k + 3)} \times \dfrac {4(k + 1)} {2(k - 2)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac { 4(k - 2)(k - 9) \times 4(k + 1)} { 2(k + 1)(k + 3) \times 2(k - 2)} $ $p = \dfrac {16(k - 2)(k - 9)(k + 1)} {4(k + 1)(k + 3)(k - 2)} $ Notice that $(k + 1)$ and $(k - 2)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac {16(k - 2)(k - 9)\cancel{(k + 1)}} {4\cancel{(k + 1)}(k + 3)(k - 2)} $ We are dividing by $k + 1$ , so $k + 1 \neq 0$ Therefore, $k \neq -1$ $p = \dfrac {16\cancel{(k - 2)}(k - 9)\cancel{(k + 1)}} {4\cancel{(k + 1)}(k + 3)\cancel{(k - 2)}} $ We are dividing by $k - 2$ , so $k - 2 \neq 0$ Therefore, $k \neq 2$ $p = \dfrac {16(k - 9)} {4(k + 3)} $ $ p = \dfrac{4(k - 9)}{k + 3}; k \neq -1; k \neq 2 $